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## G = A4×C22×C4order 192 = 26·3

### Direct product of C22×C4 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C22×C4
 Chief series C1 — C22 — C23 — C2×A4 — C22×A4 — C23×A4 — A4×C22×C4
 Lower central C22 — A4×C22×C4
 Upper central C1 — C22×C4

Generators and relations for A4×C22×C4
G = < a,b,c,d,e,f | a2=b2=c4=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 816 in 317 conjugacy classes, 81 normal (12 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C2×C4, C2×C4, C23, C23, C23, C12, A4, C2×C6, C22×C4, C22×C4, C22×C4, C24, C24, C2×C12, C2×A4, C2×A4, C22×C6, C23×C4, C23×C4, C25, C4×A4, C22×C12, C22×A4, C24×C4, C2×C4×A4, C23×A4, A4×C22×C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, A4, C2×C6, C22×C4, C2×C12, C2×A4, C22×C6, C4×A4, C22×C12, C22×A4, C2×C4×A4, C23×A4, A4×C22×C4

Smallest permutation representation of A4×C22×C4
On 48 points
Generators in S48
(1 31)(2 32)(3 29)(4 30)(5 38)(6 39)(7 40)(8 37)(9 35)(10 36)(11 33)(12 34)(13 28)(14 25)(15 26)(16 27)(17 43)(18 44)(19 41)(20 42)(21 47)(22 48)(23 45)(24 46)
(1 11)(2 12)(3 9)(4 10)(5 27)(6 28)(7 25)(8 26)(13 39)(14 40)(15 37)(16 38)(17 23)(18 24)(19 21)(20 22)(29 35)(30 36)(31 33)(32 34)(41 47)(42 48)(43 45)(44 46)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)
(5 27)(6 28)(7 25)(8 26)(13 39)(14 40)(15 37)(16 38)(17 23)(18 24)(19 21)(20 22)(41 47)(42 48)(43 45)(44 46)
(1 11)(2 12)(3 9)(4 10)(17 23)(18 24)(19 21)(20 22)(29 35)(30 36)(31 33)(32 34)(41 47)(42 48)(43 45)(44 46)
(1 5 19)(2 6 20)(3 7 17)(4 8 18)(9 25 23)(10 26 24)(11 27 21)(12 28 22)(13 48 34)(14 45 35)(15 46 36)(16 47 33)(29 40 43)(30 37 44)(31 38 41)(32 39 42)

G:=sub<Sym(48)| (1,31)(2,32)(3,29)(4,30)(5,38)(6,39)(7,40)(8,37)(9,35)(10,36)(11,33)(12,34)(13,28)(14,25)(15,26)(16,27)(17,43)(18,44)(19,41)(20,42)(21,47)(22,48)(23,45)(24,46), (1,11)(2,12)(3,9)(4,10)(5,27)(6,28)(7,25)(8,26)(13,39)(14,40)(15,37)(16,38)(17,23)(18,24)(19,21)(20,22)(29,35)(30,36)(31,33)(32,34)(41,47)(42,48)(43,45)(44,46), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (5,27)(6,28)(7,25)(8,26)(13,39)(14,40)(15,37)(16,38)(17,23)(18,24)(19,21)(20,22)(41,47)(42,48)(43,45)(44,46), (1,11)(2,12)(3,9)(4,10)(17,23)(18,24)(19,21)(20,22)(29,35)(30,36)(31,33)(32,34)(41,47)(42,48)(43,45)(44,46), (1,5,19)(2,6,20)(3,7,17)(4,8,18)(9,25,23)(10,26,24)(11,27,21)(12,28,22)(13,48,34)(14,45,35)(15,46,36)(16,47,33)(29,40,43)(30,37,44)(31,38,41)(32,39,42)>;

G:=Group( (1,31)(2,32)(3,29)(4,30)(5,38)(6,39)(7,40)(8,37)(9,35)(10,36)(11,33)(12,34)(13,28)(14,25)(15,26)(16,27)(17,43)(18,44)(19,41)(20,42)(21,47)(22,48)(23,45)(24,46), (1,11)(2,12)(3,9)(4,10)(5,27)(6,28)(7,25)(8,26)(13,39)(14,40)(15,37)(16,38)(17,23)(18,24)(19,21)(20,22)(29,35)(30,36)(31,33)(32,34)(41,47)(42,48)(43,45)(44,46), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (5,27)(6,28)(7,25)(8,26)(13,39)(14,40)(15,37)(16,38)(17,23)(18,24)(19,21)(20,22)(41,47)(42,48)(43,45)(44,46), (1,11)(2,12)(3,9)(4,10)(17,23)(18,24)(19,21)(20,22)(29,35)(30,36)(31,33)(32,34)(41,47)(42,48)(43,45)(44,46), (1,5,19)(2,6,20)(3,7,17)(4,8,18)(9,25,23)(10,26,24)(11,27,21)(12,28,22)(13,48,34)(14,45,35)(15,46,36)(16,47,33)(29,40,43)(30,37,44)(31,38,41)(32,39,42) );

G=PermutationGroup([[(1,31),(2,32),(3,29),(4,30),(5,38),(6,39),(7,40),(8,37),(9,35),(10,36),(11,33),(12,34),(13,28),(14,25),(15,26),(16,27),(17,43),(18,44),(19,41),(20,42),(21,47),(22,48),(23,45),(24,46)], [(1,11),(2,12),(3,9),(4,10),(5,27),(6,28),(7,25),(8,26),(13,39),(14,40),(15,37),(16,38),(17,23),(18,24),(19,21),(20,22),(29,35),(30,36),(31,33),(32,34),(41,47),(42,48),(43,45),(44,46)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(5,27),(6,28),(7,25),(8,26),(13,39),(14,40),(15,37),(16,38),(17,23),(18,24),(19,21),(20,22),(41,47),(42,48),(43,45),(44,46)], [(1,11),(2,12),(3,9),(4,10),(17,23),(18,24),(19,21),(20,22),(29,35),(30,36),(31,33),(32,34),(41,47),(42,48),(43,45),(44,46)], [(1,5,19),(2,6,20),(3,7,17),(4,8,18),(9,25,23),(10,26,24),(11,27,21),(12,28,22),(13,48,34),(14,45,35),(15,46,36),(16,47,33),(29,40,43),(30,37,44),(31,38,41),(32,39,42)]])

64 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 4A ··· 4H 4I ··· 4P 6A ··· 6N 12A ··· 12P order 1 2 ··· 2 2 ··· 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 3 ··· 3 4 4 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + + + + + image C1 C2 C2 C3 C4 C6 C6 C12 A4 C2×A4 C2×A4 C4×A4 kernel A4×C22×C4 C2×C4×A4 C23×A4 C24×C4 C22×A4 C23×C4 C25 C24 C22×C4 C2×C4 C23 C22 # reps 1 6 1 2 8 12 2 16 1 6 1 8

Matrix representation of A4×C22×C4 in GL5(𝔽13)

 12 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 8 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 12 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 1 1 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 12 12 11 0 0 0 0 1

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[8,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,12,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,1,0,0,0,12,1,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,11,1] >;

A4×C22×C4 in GAP, Magma, Sage, TeX

A_4\times C_2^2\times C_4
% in TeX

G:=Group("A4xC2^2xC4");
// GroupNames label

G:=SmallGroup(192,1496);
// by ID

G=gap.SmallGroup(192,1496);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,2,142,530,909]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=d^2=e^2=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations

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